Research Our research efforts strive to develop and exploit advanced mathematical methods (algebraic methods) as a means of attacking problems of current interest in Physical Chemistry. Among the problems being attacked at the present time are:

The Study of Non-Rigid Molecules During the past decade, Lie algebraic methods, introduced by our group in the early 1980’s, have been exploited to quantitatively analyze molecular spectra for a variety of rigid species. However, many cases exist, especially among extended systems, where the nuclear framework undergoes large-amplitude motion. These non-rigid systems are very difficult to treat with conventional methods based on the solution of the Schrödinger equation. Recently, we have introduced a novel algebraic scheme designed to tackle such problems and have used it to describe states of some non-rigid molecules (fulminic acid HCNO/DCNO, magnesium hydroxide MgOH/MgOD). A host of applications is possible and we intend to exploit the method to analyze a variety of molecular species.

The Study of Quantum Phase Transitions in Molecules Phase transitions are a pervasive phenomenon that occurs in all branches of physics and chemistry. Recently a new concept has been introduced into physical sciences, called quantum phase transitions. These are phase transitions that occur as a function of a coupling parameter rather than of temperature. Algebraic methods are particularly well suited to study quantum phase transitions. We have used them to study phase transitions between different molecular configurations, in particular the linear to bent transition in tri-atomic molecules and the linear to cis-bent and trans-bent transition in four-atomic molecules shown in the figure. Very recently, our group has developed the concept of excited state quantum phase transitions by means of which phase transitions both as a function of a coupling parameter and the temperature can be studied. We are applying this concept to the study of phase transitions between different molecular configurations that occur as a function of excitation energy (temperature). For example, the water molecule, H2O, undergoes a bent to linear transition at ~11,000 cm-1. An important aspect of this study is the understanding of finite-size scaling, since molecular systems are finite in size.

The Study of Intensities of Franck-Condon Transitions in Polyatomic Molecules The quantification of vibronic band intensities in Franck-Condon transitions remains a difficult and formidable task. We have analyzed, within the framework of algebraic methods, the emission spectrum of methinophosphide (HCP), connecting states with different symmetry (bent-from-linear Ã←X transition). We intend to continue this research, and extend it to transitions to the continuum.

The Study of Finite Polymer Chains An analysis of infrared and Raman intensities in finite polymer chains was initiated years ago. This analysis is particularly important for understanding how molecular aggregates are formed, going from momomer to dimer, trimer, etc. A preliminary study of the paraffins, CH3-(CH2)n-2-CH3 has been performed. Finite number (n) effects and end effects are of particular interest. We intend to extend these studies to more complex geometric structures, such as helicoidal structures, and to more than one dimension (membranes), in view of possible applications of the algebraic method to biological systems.

Although sophisticated mathematical methods are used, the research is aimed at the interpretation and understanding of experimental data, often obtained at the research laboratories of the Yale Chemistry Department. Such is the case of the study of intensities of Franck-Condon transitions, studied in collaboration with the group of P.H. Vaccaro. This research is also part of a wider interdisciplinary research that covers areas both in Physics and Chemistry. Only that part of the research that is devoted to problems in Physical Chemistry is outlined here.